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Question

# In a plane there are 37 straight lines, of which 13 pass through the point A and 11 pass through the point B. Besides, no three lines pass through one point, no lines passes through both points A and B, and no two are parallel, then the number of intersection points the lines have is equal to

A
535
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B
601
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C
728
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D
963
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Solution

## The correct option is A 535 Let each line that passes through point A be known as an A line. Let each line that passes through point B be known as a B line. Let each line that passes through neither point A nor point B be known as an N line. Since there 13 A lines, 11 B lines, and a total of 37 lines, the number of N lines =37−13−11=13 Case 1: An A line intersects with a B line Number of options for the A line =13 Number of options for the B line =11 To combine these options, we multiply: 13×11=143 Case 2: An N line intersects with an A or B line Number options for the N line =13 Number of options for the A or B line =13+11=24 To combine these options, we multiply: 13×24=312 Case 3: An N line intersects with another N line Each PAIR of N lines will yield an intersection. From the 13 N lines, the number of ways to choose2=13C2=(13×12)/(2×1)=78 Case 4: Points A and B Points A and B constitute 2 more intersections =2 Total intersections =143+312+78+2=535

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